The ascending sequence of reduced rational numbers between 0 and 1 whose denominators do not exceed a given positive integer N, arranged in increasing order and including 0/1 and 1/1.

Principle

Adjacent terms a/b and c/d in the Farey sequence of order N satisfy bc − ad = 1; mediants and the Farey addition rule generate intermediate fractions under denominator bounds.

Demonstration

For N = 5 the Farey sequence F5 is 0/1, 1/5, 1/4, 1/3, 2/5, 1/2, 3/5, 2/3, 3/4, 4/5, 1/1; adjacent pairs satisfy the determinant condition and mediants with denominator ≤5 appear between neighbors.

Misapplication

Assuming that the mediant of any two fractions always belongs to the same Farey sequence without checking the denominator bound or adjacency can produce a fraction not in the sequence.

Consequence

Farey sequences organize rationals by denominator complexity, yield identities used in modular and analytic number theory, and connect to geometric constructions (e.g., Ford circles) and continued fractions.

Reversal

The Stern–Brocot tree produces all positive rationals without a fixed denominator bound by iterated mediants, contrasting with Farey sequences that fix a maximum denominator and list only reduced fractions in an interval.

Boundary

Definition is specific to the interval [0,1] and to denominators ≤ N; one may extend definitions to other intervals or to reduced rationals without an upper denominator bound, but then properties like adjacency determinants change.

Semantic Tension

Distinguish Farey sequences from Stern–Brocot enumerations and from sorted lists of rationals by denominator alone; Farey emphasizes adjacency and the bc−ad=1 condition under a denominator cap.

Synthesis

A Farey sequence of order N is the ordered list of reduced rationals in [0,1] with denominators at most N, characterized by the unit determinant condition for neighbors and by mediant generation constrained by denominator limits.